Integral operator Riccati equations arising in stochastic Volterra control problems

We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 ($mu$ $otimes$ $mu$) for certain signed matrix measures $mu$ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.